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March 2009 Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group
Steven N. Evans
Ann. Probab. 37(2): 726-741 (March 2009). DOI: 10.1214/08-AOP418

Abstract

We consider the asymptotic behavior as n→∞ of the spectra of random matrices of the form $$\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_{n}\bigl ((k,k+1)\bigr),$$ where for each n the random variables Znk are i.i.d. standard Gaussian and the matrices ρn((k, k+1)) are obtained by applying an irreducible unitary representation ρn of the symmetric group on {1, 2, …, n} to the transposition (k, k+1) that interchanges k and k+1 [thus, ρn((k, k+1)) is both unitary and self-adjoint, with all eigenvalues either +1 or −1]. Irreducible representations of the symmetric group on {1, 2, …, n} are indexed by partitions λn of n. A consequence of the results we establish is that if λn,1λn,2≥⋯≥0 is the partition of n corresponding to ρn, μn,1μn,2≥⋯≥0 is the corresponding conjugate partition of n (i.e., the Young diagram of μn is the transpose of the Young diagram of λn), limn→∞λn,i/n=pi for each i≥1, and limn→∞μn,j/n=qj for each j≥1, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean θZ and variance 1−θ2, where θ is the constant ∑ipi2−∑jqj2 and Z is a standard Gaussian random variable.

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Steven N. Evans. "Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group." Ann. Probab. 37 (2) 726 - 741, March 2009. https://doi.org/10.1214/08-AOP418

Information

Published: March 2009
First available in Project Euclid: 30 April 2009

zbMATH: 1168.15017
MathSciNet: MR2510022
Digital Object Identifier: 10.1214/08-AOP418

Subjects:
Primary: 15A52, 60F99
Secondary: 20C30

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.37 • No. 2 • March 2009
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